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Exponential Barycenters of the Canonical Cartan Connection and Invariant Means on Lie Groups

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Matrix Information Geometry

Abstract

When performing statistics on elements of sets that possess a particular geometric structure, it is desirable to respect this structure. For instance in a Lie group, it would be judicious to have a notion of a mean which is stable by the group operations (composition and inversion). Such a property is ensured for Riemannian center of mass in Lie groups endowed with a bi-invariant Riemannian metric, like compact Lie groups (e.g. rotations). However, bi-invariant Riemannian metrics do not exist for most non compact and non-commutative Lie groups. This is the case in particular for rigid-body transformations in any dimension greater than one, which form the most simple Lie group involved in biomedical image registration. In this chapter, we propose to replace the Riemannian metric by an affine connection structure on the group. We show that the canonical Cartan connections of a connected Lie group provides group geodesics which are completely consistent with the composition and inversion. With such a non-metric structure, the mean cannot be defined by minimizing the variance as in Riemannian Manifolds. However, the characterization of the mean as an exponential barycenter gives us an implicit definition of the mean using a general barycentric equation. Thanks to the properties of the canonical Cartan connection, this mean is naturally bi-invariant. We show the local existence and uniqueness of the invariant mean when the dispersion of the data is small enough. We also propose an iterative fixed point algorithm and demonstrate that the convergence to the invariant mean is at least linear. In the case of rigid-body transformations, we give a simple criterion for the global existence and uniqueness of the bi-invariant mean, which happens to be the same as for rotations. We also give closed forms for the bi-invariant mean in a number of simple but instructive cases, including 2D rigid transformations. For general linear transformations, we show that the bi-invariant mean is a generalization of the (scalar) geometric mean, since the determinant of the bi-invariant mean is the geometric mean of the determinants of the data. Finally, we extend the theory to higher order moments, in particular with the covariance which can be used to define a local bi-invariant Mahalanobis distance.

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References

  1. Absil, P.A., Mahony, R., Sepulchre, R.: Optimization Algorithms on Matrix Manifolds. Princeton University Press, Princeton (2008)

    Google Scholar 

  2. Adler, R.L., Dedieu, J.-P., Margulies, J.Y., Martens, M., Shub, M.: Newton’s method on Riemannian manifolds and a geometric model for the human spine. IMA J. Numer. Anal. 22(3), 359–390 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  3. Afsari, B.: Riemannian \(l^p\) center of mass: existence, uniqueness, and convexity. Proc. AMS 180(2), 655–673 (2010)

    MathSciNet  Google Scholar 

  4. Arnaudon, M.: Espérances conditionnelles et \(C\)-martingales dans les variétés. In: Yor, M., Azema, J., Meyer, P.A. (eds.) Séminaire De Probabilités XXVIII. Lecture Notes in Math., vol. 1583, pp. 300–311. Springer, Berlin (1994)

    Google Scholar 

  5. Arnaudon, M.: Barycentres convexes et approximations des martingales continues dans les variétés. In: Yor, M., Azema, J., Meyer, P.A. (eds.) Séminaire de probabilités XXIX. Lecture Notes in Math., vol. 1613, pp. 70–85. Springer, Berlin (1995)

    Google Scholar 

  6. Arnaudon, M., Li, X.-M.: Barycenters of measures transported by stochastic flows. Ann. Probab. 33(4), 1509–1543 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  7. Arsigny, V.: Processing data in Lie groups: an algebraic approach. Application to non-linear registration and diffusion tensor MRI. Thèse de Sciences (Ph.D. Thesis), École polytechnique (2006)

    Google Scholar 

  8. Arsigny, V., Pennec, X., Ayache, N.: Bi-invariant means in Lie groups. Application to left-invariant polyaffine transformations. Research Report rr-5885, INRIA Sophia-Antipolis (2006)

    Google Scholar 

  9. Banach, S.: Sur les opérations dans les ensembles abstraits et leur application aux équations intégrales. Fund. Math. 3, 133–181 (1922)

    MATH  Google Scholar 

  10. Basser, P.J., Mattiello, J., Le Bihan, D.: MR diffusion tensor spectroscopy and imaging. Biophys. J. 66, 259–267 (1994)

    Article  Google Scholar 

  11. Bhattacharya, R., Patrangenaru, V.: Nonparametric estimation of location and dispersion on Riemannian manifolds. J. Stat. Plan. Inference 108, 23–36 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  12. Bhattacharya, R., Patrangenaru, V.: Large sample theory of intrinsic and extrinsic sample means on manifolds, I. Ann. Stat. 31(1), 1–29 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  13. Bhattacharya, R., Patrangenaru, V.: Large sample theory of intrinsic and extrinsic sample means on manifolds, II. Ann. Stat. 33(3), 1225–1259 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  14. Boisvert, J., Cheriet, F., Pennec, X., Labelle, H., Ayache, N.: Geometric variability of the scoliotic spine using statistics on articulated shape models. IEEE Trans. Med. Imaging 27(4), 557–568 (2008)

    Article  Google Scholar 

  15. Burbea, J., Rao, C.R.: Entropy differential metric, distance and divergence measures in probability spaces: a unified approach. J. Multivariate Anal. 12, 575–596 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  16. Calvo, M., Oller, J.M.: An explicit solution of information geodesic equations for the multivariate normal model. Stat. Dec. 9, 119–138 (1991)

    MathSciNet  MATH  Google Scholar 

  17. Cartan, E., Schouten, J.A.: On the geometry of the group-manifold of simple and semi-simple groups. Proc. Akad. Wekensch (Amsterdam) 29, 803–815 (1926)

    Google Scholar 

  18. Cheng, S.H., Higham, N.J., Kenney, C.S., Laub, A.J.: Approximating the logarithm of a matrix to specified accuracy. SIAM J. Matrix Anal. Appl. 22(4), 1112–1125 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  19. Corcuera, J.M., Kendall, W.S.: Riemannian barycentres and geodesic convexity. Math. Proc. Camb. Phil. Soc. 127, 253–269 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  20. Thompson, D.W.: On Growth and Form. Cambridge University Press, Cambridge (1917)

    Google Scholar 

  21. Dedieu, J.P., Priouret, P., Malajovich, G.: Newton’s method on Riemannian manifolds: covariant alpha theory. IMA J. Numer. Anal. 23(3), 395–419 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  22. do Carmo, M.: Riemannian Geometry. Mathematics. Birkhäuser, Boston (1992)

    Google Scholar 

  23. Emery, M., Mokobodzki, G.: Sur le barycentre d’une probabilité dans une variété. In: Yor, M., Azema, J., Meyer, P.A. (eds.) Séminaire De Probabilités XXV. Lecture Notes in Math., vol. 1485, pp. 220–233. Springer, Berlin (1991)

    Google Scholar 

  24. Fillard, P., Arsigny, V., Pennec, X., Hayashi, K.M., Thompson, P.M., Ayache, N.: Measuring brain variability by extrapolating sparse tensor fields measured on sulcal lines. Neuroimage 34(2), 639–650 (2007)

    Article  Google Scholar 

  25. Fletcher, P.T., Lu, C., Joshi, S.: Statistics of shape via principal component analysis on Lie groups. In: Proceedings of International Conference on Computer Vision and Pattern recognition (CVPR’2003), vol. 1, pp. 95–101. IEEE Computer Society, Madison (2003)

    Google Scholar 

  26. Fletcher, P.T., Lun, C., Pizer, S.M., Joshi, S.: Principal geodesic analysis for the study of nonlinear statistics of shape. IEEE Trans. Med. Imaging 23(8):995–1005 (2004)

    Google Scholar 

  27. Fréchet, M.: Les éléments aléatoires de nature quelconque dans un espace distancié. Ann. Inst . Henri Poincaré 10, 215–310 (1948)

    Google Scholar 

  28. Gallier, J.: Logarithms and square roots of real matrices. Technical Reports (cis), UPENN (2008). arXiv:0805.0245

    Google Scholar 

  29. Gallot, S., Hulin, D., Lafontaine, J., Riemannian Geometry, 2nd edn. Springer, Berlin (1993)

    Google Scholar 

  30. Godement, R.: Introduction à la Théorie des Groupes de Lie, Tomes I et II. Publications Mathématiques de l’Université Paris VII (1982)

    Google Scholar 

  31. Grenander, U.: General Pattern Theory: A Mathematical Study of Regular Structures. Oxford University Press, New York (1993)

    Google Scholar 

  32. Groisser, D.: Newton’s method, zeroes of vector fields, and the Riemannian center of mass. Adv. Appl. Math. 33, 95–135 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  33. Hall, B.C.: Lie Groups, Lie Algebras, and Representations: An Elementary Introduction. Graduate Texts in Mathematics, vol. 222. Springer, Berlin (2003)

    Google Scholar 

  34. Helgason, S: Differential Geometry, Lie Groups, and Symmetric Spaces. Academic Press, New York (1978)

    Google Scholar 

  35. Higham, N.J.: The scaling and squaring method for the matrix exponential revisited. SIAM J. Matrix Anal. Appl. 26(4), 1179–1193 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  36. Horn, R.A., Johnson, C.R.: Matrix Analysis. Cambridge University Press, Cambridge (1990)

    Google Scholar 

  37. Karcher, H.: Riemannian center of mass and mollifier smoothing. Commun. Pure Appl. Math. 30, 509–541 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  38. Kendall, W.S.: Probability, convexity, and harmonic maps with small image I: uniqueness and fine existence. Proc. Lond. Math. Soc. 61(2), 371–406 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  39. Kenney, C., Laub, A.J.: Condition estimates for matrix functions. SIAM J. Matrix Anal. Appl. 10, 191–209 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  40. Khesin, B.A., Wendt, R.: The Geometry of Infinite Dimensional Lie groups. Ergebnisse der Mathematik und ihrer Grenzgebiete, vol. 51. 3. Folge/A Series of Modern Surveys in Mathematics. Springer, Berlin (2009)

    Google Scholar 

  41. Klingenberg, W.: Riemannian Geometry. Walter de Gruyter, Berlin (1982)

    MATH  Google Scholar 

  42. Lang, S.: Algebra. Graduate Texts in Mathematics, 3rd rev. edn. Springer, Berlin (2002; corr. 4th printing edition, 2004)

    Google Scholar 

  43. Laquer, T.: Invariant affine connections on Lie groups. Trans. Am. Math. Soc. 331(2), 541–551 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  44. Le, H.: Locating Fréchet means with application to shape spaces. Adv. Appl. Probab. 33, 324–338 (2001)

    Article  MATH  Google Scholar 

  45. Le, H.: Estimation of Riemannian barycenters. LMS J. Comput. Math. 7, 193–200 (2004)

    MathSciNet  MATH  Google Scholar 

  46. Le Bihan, D., Mangin, J.-F., Poupon, C., Clark, C.A., Pappata, S., Molko, N., Chabriat, H.: Diffusion tensor imaging: concepts and applications. J. Mag. Reson. Imaging 13(4), 534–546 (2001)

    Article  Google Scholar 

  47. Mahony, R., Manton, R.: The geometry of the Newton method on non-compact Lie groups. J. Global Opt. 23, 309–327 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  48. Miller, M.I., Younes, L.: Group actions, homeomorphisms, and matching: a general framework. Int. J. Comput. Vis. 41(1/2), 61–84 (2001)

    Article  MATH  Google Scholar 

  49. Moakher, M.: Means and averaging in the group of rotations. SIAM J. Matrix Anal. Appl. 24(1), 1–16 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  50. Oller, J.M., Corcuera, J.M.: Intrinsic analysis of statistical estimation. Ann. Stat. 23(5), 1562–1581 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  51. Owren, B., Welfert, B.: The Newton iteration on Lie groups. BIT Numer. Math. 40(1), 121–145 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  52. Peeters, R.L.M., Hanzon, B.: The Riemannian interpretation of Gauss-Newton and scoring, with application to system identification. FEWEB Research Memoranda 1992-22, Vrije Universiteit Amsterdam (1992)

    Google Scholar 

  53. Pennec, X.: L’incertitude dans les problèmes de reconnaissance et de recalage—applications en imagerie médicale et biologie moléculaire. Thèse de Sciences (Ph.D. Thesis), Ecole Polytechnique, Palaiseau (France) (1996)

    Google Scholar 

  54. Pennec, X.: Computing the mean of geometric features—application to the mean rotation. Research Report RR-3371, INRIA (1998)

    Google Scholar 

  55. Pennec, X.: Intrinsic statistics on Riemannian manifolds: basic tools for geometric measurements. J. Math. Imaging Vis. 25(1), 127–154 (2006)

    Article  MathSciNet  Google Scholar 

  56. Pennec, X.: Statistical Computing on Manifolds for Computational Anatomy. Habilitation à diriger des Recherches. Nice Sophia-Antipolis Univ. (2006)

    Google Scholar 

  57. Pennec, X., Fillard, P., Ayache, N.: A Riemannian framework for tensor computing. Int. J. Comput. Vis. 66(1), 41–66 (2006)

    Article  MathSciNet  Google Scholar 

  58. Postnikov, M.M.: Geometry VI: Riemannian Geometry. Encyclopedia of Mathematical Science. Springer, Berlin (2001)

    Google Scholar 

  59. Samelson, H.: On the Brouwer fixed point theorem. Portugal. Math. 22, 264–268 (1963)

    MathSciNet  Google Scholar 

  60. Sattinger, D.H., Weaver, O.L.: Lie Groups and Algebras with Applications to Physics, Geometry, and Mechanics. AMS, vol. 61. Springer, New York (1986)

    Google Scholar 

  61. Skovgaard, L.T.: A Riemannian geometry of the multivariate normal model. Scand. J. Statistics 11, 211–223 (1984)

    MathSciNet  MATH  Google Scholar 

  62. Smith, S.T.: Optimization techniques on Riemannian manifolds, Hamiltonian and gradient flows, algorithms and control. Fields Inst. Commun. 3, 113136 (1994)

    Google Scholar 

  63. Sternberg, S.: Lectures on Differential Geometry. Prentice Hall Mathematics Series. Prentice Hall, Englewood Cliffs (1964)

    Google Scholar 

  64. Trouvé, A.: Diffeomorphisms groups and pattern matching in image analysis. Int. J. Comput. Vis. 28(3), 213–221 (1998)

    Article  Google Scholar 

  65. Udriste, C.: Convex functions and optimization methods on Riemannian manifolds. Mathematics and its Applications, vol. 297. Kluwer, Dordrecht (1994)

    Google Scholar 

  66. Woods, R.P.: Characterizing volume and surface deformations in an atlas framework: theory, applications, and implementation. Neuroimage 18(3), 769–88 (2003)

    Article  Google Scholar 

  67. Wüstner, M.: A connected Lie group equals the square of the exponential image. J. Lie Theory 13, 307–309 (2003)

    MathSciNet  MATH  Google Scholar 

  68. Yang, L.: Riemannian median and its estimation. LMS J. Comput. Math. 13, 461–479 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  69. Yang, L.: Medians of probability measures in Riemannian manifolds and applications to radar target detection. Ph.D. Thesis, Poitier University (2011)

    Google Scholar 

  70. Younes, L.: Shapes and Diffeomorphisms. Applied Mathematical Sciences, vol. 171. Springer, Berlin (2011)

    Google Scholar 

  71. Zefran, M., Kumar, V., Christopher, C.: Metrics and connections for rigid-body kinematics. Int. J. Robot. Res. 18(2), 243–258 (1999)

    Article  Google Scholar 

  72. Ziezold, H.: On expected figures in the plane. In: Hübler, A., Nagel, W., Ripley, B.D., Werner, G. (eds.) Geobild ’89. Mathematical Research, vol. 51, pp. 105–110. Akademie, Berlin (1989)

    Google Scholar 

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Acknowledgments

The authors would like to thanks the reviewers for their very detailed and insightful comments which considerably improved the manuscript. However, as mentioned in the text, some of their questions remains open for future investigations.

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  1. Asclepios Project-Team, INRIA Sophia-Antipolis Méditerranée, 2004 Route des Lucioles BP93, 06902, Sophia Antipolis Cedex, France

    Xavier Pennec & Vincent Arsigny

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  1. Xavier Pennec
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Correspondence to Xavier Pennec .

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  1. Higashi Gotanda 3-14-13, Shinagawa-Ku, 141-0022, Japan

    Frank Nielsen

  2. S.J.S. Sansanwal Marg 7, Delhi, 110016, India

    Rajendra Bhatia

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Pennec, X., Arsigny, V. (2013). Exponential Barycenters of the Canonical Cartan Connection and Invariant Means on Lie Groups. In: Nielsen, F., Bhatia, R. (eds) Matrix Information Geometry. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-30232-9_7

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  • DOI: https://doi.org/10.1007/978-3-642-30232-9_7

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